63 research outputs found
Finite rigid sets in curve complexes
We prove that curve complexes of surfaces are finitely rigid: for every
orientable surface S of finite topological type, we identify a finite
subcomplex X of the curve complex C(S) such that every locally injective
simplicial map from X into C(S) is the restriction of an element of Aut(C(S)),
unique up to the (finite) point-wise stabilizer of X in Aut(C(S)). Furthermore,
if S is not a twice-punctured torus, then we can replace Aut(C(S)) in this
statement with the extended mapping class group.Comment: 19 pages, 12 figures. v2: small additions to improve exposition. v3:
conclusion of Lemma 2.5 weakened, and proof of Theorem 3.1 adjusted
accordingly. Main theorem remains unchange
Injections of mapping class groups
We construct new monomorphisms between mapping class groups of surfaces. The
first family of examples injects the mapping class group of a closed surface
into that of a different closed surface. The second family of examples are
defined on mapping class groups of once-punctured surfaces and have quite
curious behaviour. For instance, some pseudo-Anosov elements are mapped to
multi-twists. Neither of these two types of phenomena were previously known to
be possible although the constructions are elementary
Uniformly exponential growth and mapping class groups of surfaces
We show that the mapping class group of an orientable finite type surface has
uniformly exponential growth, as well as various closely related groups. This
provides further evidence that mapping class groups may be linear.Comment: 6 pages, no figure
N=2 Gauge Theories: Congruence Subgroups, Coset Graphs and Modular Surfaces
We establish a correspondence between generalized quiver gauge theories in
four dimensions and congruence subgroups of the modular group, hinging upon the
trivalent graphs which arise in both. The gauge theories and the graphs are
enumerated and their numbers are compared. The correspondence is particularly
striking for genus zero torsion-free congruence subgroups as exemplified by
those which arise in Moonshine. We analyze in detail the case of index 24,
where modular elliptic K3 surfaces emerge: here, the elliptic j-invariants can
be recast as dessins d'enfant which dictate the Seiberg-Witten curves.Comment: 42+1 pages, 5 figures; various helpful comments incorporate
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